Finite Group Databasebeta

← #30642059

Subgroups — #30642059

Order 16256.

1116 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 169 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×D508C211yesmaximal, characteristic
2C2×#16861194C212yesmaximal, complemented
3C2×C22⋊C4×D127C211yesmaximal, characteristic, complemented
4#22549794C212yesmaximal, complemented
5#22549793C212yesmaximal, complemented
6#22549754C212yesmaximal, complemented
7#22549749C211yesmaximal, characteristic, complemented
8#22549731C211yesmaximal, characteristic, complemented
9#22549731C211yesmaximal, characteristic, complemented
10#22549731C211yesmaximal, characteristic, complemented
11#385×C127C211yesmaximal, characteristic, complemented
12C24×D127C2211yescharacteristic
13C24×D127C2211yescharacteristic
14C22×D508—24no—
15C22×D508—28no—
16C22×D508C2212yes—
17C22×D508C2212yes—
18C22×C4×D127C2211yescharacteristic, complemented
19C22×C4×D127C2211yescharacteristic, complemented
20#16861194—28no—
21C2×Dic127⋊C4C2212yescomplemented
22C2×C22⋊Dic127C2211yescharacteristic
23C2×C22⋊Dic127C2211yescharacteristic
24C2×(C2×D127)⋊C4C2211yescharacteristic, complemented
25C2×(C2×D127)⋊C4C2212yescomplemented
26C2×(C2×D127)⋊C4C2211yescharacteristic, complemented
27C2×(C2×D127)⋊C4C2212yescomplemented
28C2×(C2×D127)⋊C4C2212yescomplemented
29C2×(C2×D127)⋊C4C2212yescomplemented
30C2×(C2×D127)⋊C4C2212yescomplemented
31C2×(C2×D127)⋊C4C2212yescomplemented
32C22⋊C4×D127—24no—
33C127⋊#95C2212yescomplemented
34C23×C4×C127C2211yescharacteristic
35C2×C4⋊C4×C127C2212yescomplemented
36C2×C22⋊C4×C127C2211yescharacteristic
37C2×C22⋊C4×C127C2211yescharacteristic
38#95×C127C2212yescomplemented
39C23×D127—24no—
40C23×D127—24no—
41C23×D127—22no—
42C23×D127—22no—
43C23×D127—22no—
44C23×D127—22no—
45C23×D127—22no—
46C23×D127—42no—
47C23×D127—42no—
48C23×D127D414yes—
49C23×D127C2311yescharacteristic
50C23×D127C2312yes—
51C23×D127C2311yescharacteristic
52C2×D508—232no—
53C2×D508—24no—
54C2×D508—24no—
55C2×D508—24no—
56C2×D508—24no—
57C2×D508—24no—
58C2×D508D418yes—
59C22×Dic127C2311yescharacteristic
60C22×Dic127C2311yescharacteristic
61C22×Dic127C2312yes—
62C2×C4×D127—24no—
63C2×C4×D127—22no—
64C2×C4×D127—22no—
65C2×C4×D127—42no—
66Dic127⋊C4—24no—
67C22⋊Dic127—22no—
68C22⋊Dic127—22no—
69(C2×D127)⋊C4—24no—
70(C2×D127)⋊C4—24no—
71(C2×D127)⋊C4—24no—
72(C2×D127)⋊C4—22no—
73(C2×D127)⋊C4—22no—
74(C2×D127)⋊C4—42no—
75(C2×D127)⋊C4—42no—
76(C2×D127)⋊C4—42no—
77C24×C127C2311yescharacteristic
78C22×C4×C127—22no—
79C22×C4×C127—24no—
80C22×C4×C127C2311yescharacteristic
81C22×C4×C127C2311yescharacteristic
82C22×C4×C127C2312yes—
83C22×C4×C127C2312yes—
84C4⋊C4×C127—24no—
85C22⋊C4×C127—22no—
86C22⋊C4×C127—22no—
87C22×D127—21no—
88C22×D127—21no—
89C22×D127—22no—
90C22×D127—28no—
91C22×D127—28no—
92C22×D127—22no—
93C22×D127—22no—
94C22×D127—22no—
95C22×D127—21no—
96C22×D127—22no—
97C22×D127—21no—
98C22×D127—22no—
99C22×D127—22no—
100C22×D127—22no—
101C22×D127—22no—
102C22×D127—28no—
103C22×D127—44no—
104C22×D127—48no—
105C22×D127—48no—
106C22×D127—48no—
107C22×D127—48no—
108C22×D127—44no—
109C22×D127—44no—
110C22×D127C2×D412yes—
111C22×D127C2×D414yes—
112C22×D127C2×D414yes—
113C22×D127C2×D412yes—
114D508—216no—
115D508—216no—
116C2×Dic127—22no—
117C2×Dic127—21no—
118C2×Dic127—21no—
119C2×Dic127—22no—
120C2×Dic127—22no—
121C2×Dic127—22no—
122C2×Dic127—22no—
123C4×D127—44no—
124C4×D127—44no—
125C23×C127—22no—
126C23×C127—22no—
127C23×C127C2×D412yes—
128C23×C127#4112yes—
129C23×C127#4112yes—
130C23×C127C2411yescharacteristic
131C2×C4×C127—28no—
132C2×C4×C127—22no—
133C2×C4×C127—22no—
134C2×C4×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127—22no—
137C2×C4×C127—22no—
138C2×C4×C127—22no—
139C2×C4×C127—22no—
140C2×C4×C127—22no—
141C2×C4×C127C2×D414yes—
142C22×C127—22no—
143C22×C127—22no—
144C22×C127—24no—
145C22×C127—24no—
146C22×C127#12314yescomplemented
147C22×C127C22×D411yescharacteristic
148C22×C127C22×D412yes—
149C22×C127C2×#4111yescharacteristic
150C22×C127C2×#4111yescharacteristic
151C22×C127H(5,𝔽2)12yes—
152C2×D127—22no—
153C2×D127—22no—
154C2×D127—24no—
155C2×D127—24no—
156C2×D127—22no—
157C2×D127—24no—
158C2×D127—22no—
159C2×D127—22no—
160C2×D127—24no—
161C2×D127—24no—
162C2×D127—416no—
163C2×D127—416no—
164C2×D127—42no—
165C2×D127—42no—
166C2×D127—42no—
167C2×D127—42no—
168C2×D127—42no—
169C2×D127—42no—
170C2×D127#12014yes—
171C4×C127—24no—
172C4×C127—41no—
173C4×C127—41no—
174C4×C127—42no—
175Dic127—41no—
176Dic127—41no—
177Dic127—42no—
178C2×C127—24no—
179C2×C127C2×#12011yescharacteristic
180C2×C127C2×#12311yescharacteristic
181C2×C127#53411yescharacteristic
182C2×C127#54612yes—
183C2×C127#54612yes—
184D127—24no—
185D127—42no—
186D127—44no—
187#2405—1271nomaximal, Sylow
188C127#240511yescharacteristic, complemented, Sylow
189C23×D4—1271no—
190C2×#123—1272no—
191C22×C22⋊C4—1271no—
192#394—1272no—
193#394—1272no—
194#386—1272no—
195#385—1271no—
196#385D12711yescharacteristic, complemented
197#379—1271no—
198#379—1271no—
199#379—1271no—
200C25—1271no—
201C25—1271no—
202C22×D4—1272no—
203C22×D4—1272no—
204C22×D4—2548no—
205C22×D4—2544no—
206C23×C4—1271no—
207C23×C4—1271no—
208C23×C4C2×D12711yescharacteristic
209#123—2548no—
210C2×C4⋊C4—1272no—
211C2×C4⋊C4C2×D12712yescomplemented
212C2×C22⋊C4—1272no—
213C2×C22⋊C4—1272no—
214C2×C22⋊C4—1272no—
215C2×C22⋊C4—1271no—
216C2×C22⋊C4—1271no—
217C2×C22⋊C4—1271no—
218C2×C22⋊C4—1271no—
219C2×C22⋊C4—1272no—
220C2×C22⋊C4—1272no—
221C2×C22⋊C4—1272no—
222C2×C22⋊C4—2544no—
223C2×C22⋊C4C2×D12711yescharacteristic
224C2×C22⋊C4C2×D12711yescharacteristic
225#95—1272no—
226#95C2×D12712yescomplemented
227C24—1271no—
228C24—1274no—
229C24—1272no—
230C24—1271no—
231C24—5082no—
232C24—5082no—
233C24—2544no—
234C24—2544no—
235C24—2542no—
236C24—2542no—
237C24—2542no—
238C24—2542no—
239C24—2542no—
240C24C22×D12711yescharacteristic
241C2×D4—1278no—
242C2×D4—2544no—
243C2×D4—2544no—
244C2×D4—2544no—
245C2×D4—2544no—
246C2×D4—25432no—
247C2×D4—2544no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—1271no—
251C22×C4—1272no—
252C22×C4—1271no—
253C22×C4—5082no—
254C22×C4—2542no—
255C22×C4—2542no—
256C22×C4—2544no—
257C22×C4C22×D12711yescharacteristic
258C22×C4C22×D12712yes—
259C22×C4C22×D12711yescharacteristic
260C22×C4C22×D12712yes—
261C4⋊C4—24no—
262C4⋊C4—2544no—
263C22⋊C4—22no—
264C22⋊C4—22no—
265C22⋊C4—5082no—
266C22⋊C4—5082no—
267C22⋊C4—5082no—
268C22⋊C4—2542no—
269C22⋊C4—2542no—
270C22⋊C4—2544no—
271C22⋊C4—2544no—
272C22⋊C4—2544no—
273C22⋊C4—2542no—
274C22⋊C4—2542no—
275C23—22no—
276C23—22no—
277C23—1274no—
278C23—1272no—
279C23—1274no—
280C23—1272no—
281C23—5084no—
282C23—5088no—
283C23—5088no—
284C23—5088no—
285C23—5088no—
286C23—5084no—
287C23—5084no—
288C23—2548no—
289C23—2548no—
290C23—2548no—
291C23—2542no—
292C23—2542no—
293C23—2542no—
294C23—2542no—
295C23—2541no—
296C23—2542no—
297C23—2541no—
298C23—2542no—
299C23—2542no—
300C23—2542no—
301C23—2541no—
302C23—2541no—
303C23—2542no—
304C23D4×D12712yes—
305C23#1452324812yes—
306C23#1452324912yes—
307C23C23×D12711yescharacteristic
308D4—25416no—
309D4—25416no—
310C2×C4—28no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—5084no—
321C2×C4—5084no—
322C2×C4—2542no—
323C2×C4—2541no—
324C2×C4—2541no—
325C2×C4—2542no—
326C2×C4—2542no—
327C2×C4—2542no—
328C2×C4—2542no—
329C2×C4D4×D12714yes—
330C22—22no—
331C22—22no—
332C22—24no—
333C22—24no—
334C22—1274no—
335C22—50816no—
336C22—50816no—
337C22—5082no—
338C22—5082no—
339C22—5082no—
340C22—5082no—
341C22—5082no—
342C22—5082no—
343C22—2544no—
344C22—2542no—
345C22—2542no—
346C22—2542no—
347C22—2544no—
348C22—2544no—
349C22—2542no—
350C22—2542no—
351C22—2544no—
352C22—2544no—
353C22#1686119414yescomplemented
354C22C2×D4×D12711yescharacteristic
355C22C2×D4×D12712yes—
356C22C2×#1452324811yescharacteristic
357C22C2×#1452324911yescharacteristic
358C22#1686125512yes—
359C4—24no—
360C4—41no—
361C4—41no—
362C4—42no—
363C4—5081no—
364C4—5081no—
365C4—5082no—
366C2—24no—
367C2—5084no—
368C2—5082no—
369C2—2544no—
370C2#120×D12711yescharacteristic
371C2C2×#1686119411yescharacteristic
372C2#2255043711yescharacteristic
373C2#2255051712yes—
374C2#2255052212yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#385×C127C2
prime radicalC23×D508C2
hypercenter#385D127
hypocenterC127#2405
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #385

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#16861194C2semidirect
3. C2×C22⋊C4×D127C2semidirect
4. #22549794C2semidirect
5. #22549793C2semidirect
6. #22549754C2semidirect
7. #22549749C2semidirect
8. #22549731C2semidirect
9. #22549731C2semidirect
10. #22549731C2semidirect
11. #385×C127C2semidirect
18. C22×C4×D127C22semidirect
19. C22×C4×D127C22semidirect
21. C2×Dic127⋊C4C22semidirect
24. C2×(C2×D127)⋊C4C22semidirect
25. C2×(C2×D127)⋊C4C22semidirect
26. C2×(C2×D127)⋊C4C22semidirect
27. C2×(C2×D127)⋊C4C22semidirect
28. C2×(C2×D127)⋊C4C22semidirect
29. C2×(C2×D127)⋊C4C22semidirect
30. C2×(C2×D127)⋊C4C22semidirect
31. C2×(C2×D127)⋊C4C22semidirect
33. C127⋊#95C22semidirect
35. C2×C4⋊C4×C127C22semidirect
38. #95×C127C22semidirect
146. C22×C127#123semidirect
188. C127#2405semidirect
196. #385D127semidirect
211. C2×C4⋊C4C2×D127semidirect
226. #95C2×D127semidirect
353. C22#16861194semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C23×D5082
C2×#168611942
C2×C22⋊C4×D1272
#225497942
#225497932
#225497542
#225497492
#225497312
#385×C1272
#2405127